We consider the task of estimating sparse discrete distributions under local differential privacy and communication constraints. Under local privacy constraints, we present a sample-optimal private-coin scheme that only sends a one-bit message per user. For communication constraints, we present a public-coin scheme based on random hashing functions, which we prove is optimal up to logarithmic factors. Our results show that the sample complexity only depends logarithmically on the ambient dimension, thus providing significant improvement in sample complexity under sparsity assumptions. Our lower bounds are based on a recently proposed chi-squared contraction method.

翻译：我们考虑的是在地方差异隐私和通信限制下估算分散分散分布的任务。在本地隐私限制下,我们提出了一个样本优化的私人煤炭计划,它只给每个用户发送一比特的信息。关于通信限制,我们提出了一个基于随机散列功能的公共煤炭计划,我们证明这个计划最符合对数因素。我们的结果显示,样本的复杂性仅仅在逻辑上取决于环境层面,从而在垃圾度假设下极大地改进了样本的复杂性。我们较低的界限是基于最近提出的奇孔缩缩缩法。